© The Authors 2023

1. Introduction

The solar chromosphere is shaped from long, dense fibrils (Rutten 2007) that connect large magnetic features organized on mesogranular to supergranular scales. These magnetic features are the centers of most chromospheric dynamics (Tsiropoula et al. 2012). Radially extending from them are dark stable features called mottles, visible in the cores of chromospheric spectral lines. Moving to the wings of chromospheric lines, features named rapid blue- and redshifted excursions (RBEs and RREs Rouppe van der Voort et al. 2009) become dominant. Unlike mottles, these features are very dynamic, short-lived, long, and thin, but also visible near the edges of the magnetic network. Although it is generally believed that they play an important role in mass-loading and heating the solar corona, they are still a puzzle.

The RBEs were first detected in the blue wing of the Ca II 854.2 nm line as a sudden broadening in λt data slices (Langangen et al. 2008). Their lifetimes, spatial extent, and location, as well as sudden disappearance, suggested that they might be the on-disk counterparts of the then newly discovered off-limb spicule dynamics (de Pontieu et al. 2007). Subsequent papers based on observations with higher spatial and temporal resolution further strengthened this hypothesis (Rouppe van der Voort et al. 2009; Sekse et al. 2012, 2013a). The spectral imaging data taken in both Ca II 854.2 nm and Hα lines revealed that the RBEs vary strongly between these lines. In the Ca II 854.2 nm lines we see significantly fewer and shorter features close to the magnetic field concentrations, and generally they appear before the Hα RBEs. Also, the corresponding Doppler shifts differ, being on the order of 20 − 50 km s⁻¹ in the Hα and 15 − 20 km s⁻¹ in the Ca II 854.2 nm line. The differences were explained by the chromospheric opacity in the lines, with Hα sampling higher layers and naturally faster upflows.

The properties of these features also differ depending on the detection method used and the temporal cadence of the data. The distribution of the RBE lifetimes ranges from 5 to 60 s, with low cadence data shifting it to the higher values. This may be due to the high recurrence times around magnetic elements, which gives apparently longer lifetimes at a lower cadence. Furthermore, the overall behavior of RBEs changes with temporal resolution. The first studies claimed that RBEs move away from magnetic concentration from the footpoint to the top with increasing Doppler shifts and widths. The higher cadence data, however, revealed many more RBEs that display erratic behavior. While there were cases that had a distinct rise phase, there were also features that showed a complicated mixture of longitudinal variations. There were many cases in which the RBEs seemingly came to a halt, retreated, or just appeared suddenly along the full length within a few seconds. Some RBEs moved sideways over a few hundred kilometers, while others were hardly displaced at all. It also became clear that RBEs appear in groups, with large and dark jet-like features appearing first followed by numerous shorter, thinner jets (Yurchyshyn et al. 2013; Samanta et al. 2019).

The RREs are similar absorption features, but in the red wings of the Ca II 854.2 nm and Hα lines (Sekse et al. 2013b; Kuridze et al. 2015; Shetye et al. 2016; Bose et al. 2019). They are found over the whole solar disk and are located in the same regions as RBEs. The lengths, widths, lifetimes, and average Doppler velocity of RREs are similar to those of RBEs, and they have similar occurrence rates, with the total occurrence rate increasing toward the limb. Most RREs and RBEs are observed in isolation, but many examples of parallel and touching RRE–RBE pairs have also been found in the same spicule. Kuridze et al. (2015) detected more RREs and Sekse et al. (2013b) more RBEs. The latter study also shows that the RRE/RBE detection count ratio increases from the disk center to the limb from 0.52 to 0.74. The higher number of RBEs and the decreased imbalance toward the limb is interpreted as an indication that field-aligned upflows make a significant contribution to the net Doppler shift of the structure.

The transverse displacements of RREs and RBEs are also similar. There are examples of transitions from an RRE to an RBE and vice versa that sometimes appear to occur along the structure. In some cases, even oscillatory behavior can be found with periods averaging 90 s and amplitudes of 200 km. All these characteristics are in line with the hypothesis that RREs and RBEs are an indication of Alfvénic waves associated with a swaying motion. Namely, the movement of the “flux tube” is detected in both image planes as transverse motion and along the line of sight as sudden blue- and redshifts in the wings of spectral lines. The formation of such phenomena would also arise naturally in the case of magnetic reconnection, which was indicated by observations that show RBEs generally appearing when new, mixed, or unipolar fields are detected in close proximity to network fields (Yurchyshyn et al. 2013; Deng et al. 2015; Samanta et al. 2019). Although this scenario explains all peculiarities of RBEs and RREs, alternative hypotheses must also be mentioned. One alternative explanation is that the observed transverse motions are not waves, but just transverse displacements of the flux tube as a whole from its initial position, as happens when the interaction timescale between the granule and the flux tube is comparable to or greater than the chromospheric cutoff period (Hasan & Kalkofen 1999). Another alternative explanation is that the sudden shifts in the spectral lines may be an indication of sheet-like structures (Judge et al. 2012; Lipartito et al. 2014).

A large number of numerical experiments have tried to explain the formation and origin of spicules (e.g. Uchida 1969; Hollweg 1982; Suematsu et al. 1982; Murawski & Zaqarashvili 2010; Jess et al. 2012; Iijima & Yokoyama 2017; Martínez-Sykora et al. 2017). However, only two studies tried to explicitly model RBEs and RREs (Kuridze et al. 2015; Srivastava et al. 2017). These models fail to reproduce many properties of RBEs and RREs. Still, they indicate that RBEs and RREs may be signatures of Alfvén waves generated by torsional driving in the photosphere. The present manuscript describes a three-dimensional (3D) model that reproduces the spatial distribution, lateral movement, length, and lifetimes of RBEs and RREs, as well as strongly blue- or redshifted asymmetric Hα line profiles. We start with identifying the RBE and RRE signatures in synthetic spectra generated from the numerical models. We then compare their properties with observations and further study their formation.

2. Model atmospheres and forward synthesis

The MURaM code (Vögler et al. 2005; Rempel 2017) generates numerical 3D models of the solar atmosphere. The model is built in phases starting from the nonmagnetic convection simulation, to which a uniform bipolar field of 200 G is added. After the field configuration evolves, the potential field extrapolations are used to extend the computational domain and include the upper atmosphere. The final extent of the simulation domain is 40 × 40 × 22 Mm, spanning vertically from −8 Mm to 14 Mm above the average τ₅₀₀ = 1 height. At the bottom boundary, a horizontal field at roughly equipartition field strength is allowed to emerge into the domain (Rempel 2014). At the upper boundary, the field is set to be vertical. Also, the upper boundary is open to outflows but closed to inflows. The horizontal boundaries are periodic. In the last phases, the model is first run at a lower resolution and then again at the double resolution until a relaxed state is achieved.

The grid spacing of the simulation we used is 39 km and 21 km in the horizontal and vertical directions, respectively. The model contains only the most basic physics needed to be considered “comprehensive”: 3D gray local thermodynamic equilibrium radiative transfer, a tabulated local thermodynamic equilibrium equation of state, Spitzer heat conduction, and optically thin radiative losses in the corona based on CHIANTI (Landi et al. 2012). The numerical diffusive and resistive terms in this run were set so that the effective magnetic Prandtl number P_m > 1. This makes viscous heating the largest contributor. The maximum Alfvén velocity was chosen as max(2c_s, 3|v_max|), where v_max is the maximum velocity and c_s the speed of sound (Rempel 2017). This gives a largest unaffected Alfvén velocity of 900 km s⁻¹ (Chen et al. 2022). The snapshots were outputted every 2.5 s, and every fourth snapshot was used as input for the subsequent radiative transfer computations.

For these computations, we used Multi3D (Leenaarts & Carlsson 2009) with the model atmosphere, which we cropped in two ways. The vertical span of the input model was limited to 360 points and included the range of temperatures relevant for the formation of the Hα line. In the horizontal direction, we used every second grid point to make computations faster. Tests show that this does not affect the appearance of synthetic features. Hα was calculated using a three-level plus continuum H model atom created by Bjørgen et al. (2019) to increase numerical stability in Multi3D. The synthetic spectra were generated for 29 snapshots in total.

3. Results

3.1. Spatial distribution of synthetic RBEs and RREs

Figure 1 shows synthetic Hα images generated from one of the snapshots. The magnetic field configuration is visible from the distribution of long fibrils in the Hα core, shown in the middle panel. Panels to the sides show synthetic RBEs and RREs. The wavelength positions correspond to Doppler shifts of ±36 km s⁻¹, similar to observations. Thin elongated dark features are visible in both Hα wings, resembling their observed counterparts. In addition to the thin features, the red wing image shows slightly more spread-out structures, especially close to the “slope” generated by the inclined magnetic field around [7, 25] Mm. The dark circular structure at [33, 20] Mm visible in both wings is a pore.

Fig. 1. Synthetic Hα images at different wavelength positions: the blue wing at Δv = −36 km s−1 (left), the nominal line center (middle), and the red wing at Δv = 36 km s−1 (right). The vertical blue line marks the position of the cut shown in Fig. 4. The movie is available online

Inspection of the movie corresponding to Fig. 1 reveals different kinds of features. In both RBE and RRE images, there are examples that show apparent motion or an increase or decrease in distance from the magnetic network. There are also many examples that appear and disappear in situ, seemingly halfway between the magnetic footpoints. Many features show transverse motion. All this agrees with findings based on observations.

To study the statistics of the synthetic RBEs and RREs, we applied simple filtering to the Hα wing images, exposing all regions where the intensity was below 22.3 × 10⁻⁶ erg cm⁻² s⁻¹ Hz⁻¹ ster⁻¹ and the aspect ratio larger than 15. These numbers were chosen somewhat arbitrarily. The final masks summed over 29 snapshots are visible in Fig. 2. The figure shows the occurrence and feature position with respect to magnetic elements. The chosen criteria allow the inclusion of fairly short features, such as the one at [3, 30] Mm in the top panel. It also allows many extended structures visible in the Hα red wing images to be counted as RREs, as the bottom panel of Fig. 2 shows. Although magnetic elements are distributed all over the simulation domain, the maps nevertheless show areas with reduced activity at [25, 2.5], [28, 15], and [35, 30] Mm. In other regions, RREs and RBEs can be found in seemingly equal numbers. The exception is visible at the base of the slope, where RREs prevail.

Fig. 2. Automatically identified synthetic RBEs and RREs. The panels show summed masks for RBEs and RREs, respectively. The black contours outline the unsigned photospheric vertical field of 1000 G.

3.2. Properties of synthetic RBEs and RREs

We clustered masked features that overlap from one snapshot to the other and counted them as one feature. This gives a total number of 231 RBEs and 322 RREs. The distribution of their lifetimes is shown in Fig. 3. Many features traced this way appear and disappear in one snapshot, giving them lifetimes of less than 10 s. Most of the extra RREs were placed into this bin. The distribution exponentially decreases with the lifetime. Only around 5% of both RREs and RBEs seem to be visible for at least a minute.

Fig. 3. Lifetime (left) and length (right) of tracked synthetic RBEs (black) and RREs (dashed red).

For each case, we tracked the evolution and included the most extended length in the distribution, shown in the right of Fig. 3. We see that most features are shorter than 3 Mm in size, although in some cases they extend to over 5 Mm. There seems to be no preference for the additional, short-lived RREs. They seem to be distributed proportionally to all length bins.

3.3. Formation of synthetic RBEs and RREs

Figure 4 shows a vertical cut over several RBEs and RREs at the time instant shown in Fig. 1. Overplotted are τ = 1 contours for the same wavelength positions as in Fig. 1. The figure shows that the Hα line core is formed in the range 2 − 4 Mm. The majority of the events, four RBEs and two RRE, are formed close to the Hα line core and at similar heights, which results in similar intensities, visible in Fig. 1. The bottom panel of Fig. 4 shows that at each position where RBEs and RREs are formed there is a strong upflow or downflow in the vertical velocity map. The field strength in those regions is still higher than 100 G, as the third panel in Fig. 4 shows.

Fig. 4. Vertical cut along the blue vertical lines shown in Fig. 1. Vertical cuts in temperature, density, magnetic field strength, and vertical velocity are shown from top to bottom. The black line in all panels shows the τ = 1 at the nominal position of the Hα line center. Green and red contours mark τ = 1 levels at wavelength positions that correspond to Doppler shifts of Δv = −36 km s−1 and Δv = 36 km s−1, respectively.

The examples at x = 10.7 Mm and x = 11.25 Mm in Fig. 4 are analyzed in more detail in Figs. 5 and 6 using four-panel line formation diagrams (Carlsson & Stein 1997; Leenaarts et al. 2010). An example of an RBE is shown in Fig. 5. The emergent profile in the lower-right panel is strongly blueshifted with a small reversal at the red wing. Overplotted in the same panel is the line-of-sight velocity profile, which displays a shift in the same direction. The predominant upflows, which reach 40 km s⁻¹, are in the height range 2 − 3 Mm. Because of these upflows, the location with high χ_ν/τ_ν (upper-left panel) is strongly shifted at z = 3 Mm, exactly where the optical depth reaches unity at those frequencies (lower-left panel). The source function is completely decoupled from the Planck function and stays fairly flat, albeit with two small local maxima higher up, at 3.5 and 4 Mm. The opacity has a shape characteristic for the Hα line with sharp, sudden shifts of τ = 1 in both wings but strongly shifted to the positive frequencies. As a result, the contribution function jumps to the height of 3 Mm and stays there over a wide range of frequencies. A small reversal in the red wing is formed due to a temperature and density increase, which is also visible in Fig. 4. The formation mechanism of the red wing reversal is the same as for the spectral signatures of Ellerman bombs (Danilovic 2017).

Fig. 5. Hα four-panel diagram for the RBE at x = 10.7 Mm in Fig. 4. The plotted quantities, as functions of frequency and height, are indicated in the upper-left corner of each subplot and plotted in a grayscale. The vertical component of velocity as a function of height is shown in all panels (solid white line), with upward velocity being positive. The red line marks the τ = 1 position as a function of frequency and height. Planck and total source function are plotted in the upper-right panel with dotted and dashed lines, respectively. The line profile is shown with a solid white line in the lower-right panel.

Fig. 6. Hα four-panel diagram for the RRE at x = 11.25 Mm in Fig. 4. The outline is the same as in Fig. 5.

The four-panel diagram created for the pixel at x = 11.25 Mm in Fig. 4 is presented in Fig. 6. This pixel experiences a much more complex opacity distribution than in the case of Fig. 5. Here, we have numerous bright regions in the upper-left panel at a variety of altitudes and frequencies. Two major effects on the contribution function come from the high χ_ν/τ_ν located at z = 3 and z = 2.3 − 2.5 Mm. The latter is the location where a strong downflow reaches 50 km s⁻¹. At the former location, there is a switch from an upflow to a downflow; both are slightly weaker, reaching only 20 km s⁻¹. The source function shows a local maximum in both locations. As a result, line wings and core intensities all experience contributions from both heights. The blueshifted material also coincides with the location where the optical depth reaches unity at that frequency, so the blue wing gets extended, resulting in a very asymmetric line profile.

3.4. Origin of synthetic RBEs and RREs

We chose another 14 examples around the center of the simulation domain to study what causes their appearance. An equal number of RBEs and RREs were chosen. They appear at different time instances distributed over nine different snapshots. Figure 7 gives an overview of the positions of one traced field line per each chosen feature. We traced the magnetic field lines as curvatures in 3D space (Leenaarts et al. 2015). The time cadence between snapshots used for the field tracing was around 2.5 s. For each of the examples, we picked several seeds along the feature, visible either in the blue or red synthetic Hα wing, to advect as tracer particles forward and backward in time. For each seed, the formation height at that specific wavelength position was taken as the z coordinate in the 3D space. The individual cases are labeled F0–F13, as shown in Fig. 7. Cases F12 and F13 are the RBE and RRE visible in Fig. 4 and are further discussed in the four-panel diagrams. Figure 7 shows the position of the RRE associated with F1, which appears close to F0 when F0 disappears. The line tracing indicates that these RREs belong to the same event since their origin can be traced to overlapping field lines¹.

Fig. 7. Overview of the RBE–RRE sample chosen for magnetic field line tracing. For each of the 14 examples, a traced magnetic field line is plotted over the vertical component of the photospheric magnetic field. The diamonds mark the position of the starting seed used for line tracing: red marks RREs and blue RBEs. Yellow stars mark the position of the case F1.

To analyze the displacement of each chosen example, we placed the ‘slit” at the x coordinate of the starting seed and then followed the y and z coordinates of the traced field line in time at that fixed x position. Figure 8 shows the resulting horizontal and vertical displacements for four cases after a linear fit is subtracted. The shifts in two directions are in phase with each other when the RBE or RRE appears and sometimes show oscillatory behavior over a short time. For example, F0 shows a sinusoidal wave pattern in the y coordinate that lasts for ∼40 s in the interval t = [70, 110] s. In the case of F9, the same lasts for ∼100 s in the interval t = [40, 150] s. In most cases, we see more erratic behavior, as is the case for F6. The four cases shown in Fig. 8 illustrate that the range of displacements stays within 200 km with amplitudes of a few tens of kilometers during the oscillatory period. The field tracing reveals two common characteristics for all cases. This is illustrated in Fig. 9 using four examples of RBEs and RREs. The figure shows velocities along a traced field line as a function of time, with the starting seeds and the signals propagating with the Alfvén speed along the loop marked. The first common characteristic for all cases is that we detect a strong flow at one of the footpoints, which perturbs the field line. This is visible in the vertical component of velocity displayed in the second column. For cases F3, F9, and F13, alternating upflows and downflows are visible close to the first footpoint. For case F0, the same happens at the second footpoint. The flow propagates along loops with an Alfvén speed of ∼300 km s⁻¹ for all inspected cases. The second common characteristic is that the vertical velocity that causes RBEs and RREs to appear in Hα line wings is always associated with the component of velocity perpendicular to the magnetic field line. This is visible in the third and fourth columns, where we see a high-velocity pattern only in v_P and nothing in v_L at the time instant in proximity to the position of the starting seed. Only in case F13 is a strong flow visible along the field line v_L, although not at the position of this specific seed.

Fig. 8. Horizontal (y, solid line) and vertical (z, dashed line) displacement at constant x for four examples of RBEs and RREs. The vertical line marks the time when the seed is “planted”. The cases F0 and F1 are RREs and F6 and F9 RBEs. The labels are the same as in Fig. 7.

Fig. 9. Evolution along a field line traced for four cases, labeled in the top-left corner. From left to right: total velocity, v, vertical velocity, vz, longitudinal velocity, vL, and perpendicular velocity, vP. Diamonds mark the location of starting seeds. The dotted lines follow a signal propagating with the Alfvén speed along the loop.

We can also use instantaneous proxies for three wave modes, slow (f_long) and fast (f_fast) magneto-acoustic waves and Alfvén waves (f_alf) (Khomenko et al. 2018; Cally 2017):

$$\begin{array}{cc}& f_{\mathrm{alf}}=\widehat{e_{\Vert }}·▽\times \mathit{v}\hfill \end{array}$$(1)

$$\begin{array}{cc}& f_{\mathrm{fast}}=▽·(\mathit{v}-\widehat{e_{\Vert }}{v}_{\Vert })\hfill \end{array}$$(2)

$$\begin{array}{cc}& f_{\mathrm{long}}=\widehat{e_{\Vert }}·▽(\mathit{v}·\widehat{e_{\Vert }}),\hfill \end{array}$$(3)

where v is the velocity vector and $\widehat{e_{\Vert }}$ the field-aligned unit vector. To visualize where the wave power for each mode is located with respect to the formation of RBEs and RREs, each quantity was convolved with a contribution function at the corresponding wavelength. The resulting maps are shown in Fig. 10.

Fig. 10. Proxies of different wave modes: Alfvén (left column), fast (middle column), and slow mode waves convolved with the contribution function at different wavelength positions: blue wing at Δv = −36 km s−1 (top row), red wing at Δv = 36 km s−1 (middle row), and nominal line center (bottom row). The blue vertical line marks the position of the cut shown in Fig. 4.

There is a significant difference between maps corresponding to the line center and the line wings. The latter are largely contaminated with the contribution from the photosphere. Only in regions where RBEs and RREs occur does the contribution function shift to higher layers, as illustrated in the previous section, and the maps show elongated features. If we compare Fig. 10 with Fig. 1, we can identify many features in the intensity. Most of them are visible only in f_alf. In some cases, weak signatures can also be traced in f_fast maps. A few cases can also be identified in all three wave proxies. Cases F12 and F13 belong to the last group of features. There also seems to be a trend that f_long and f_fast signatures are located closer to the footpoints and that the f_alf proxy extends farther along a feature (see also Danilovic 2022).

4. Discussion and conclusions

Hα line profiles generated from a realistic 3D model of a solar plage show RBE and RRE signatures. Their spatial distribution, length, and lifetimes are very similar to the properties of their observed counterparts. The synthetic features appear close to the magnetic network or at the apex of the loops. Similarly to the observations, their behavior can be either consistent or erratic. In the first case, they move farther from or closer to the network. In the second case, they appear and disappear far from any magnetic concentration. They show lateral motion of a few tens to a few hundred kilometers. This agrees with observations (Sekse et al. 2012, 2013a). We identify cases where features exhibit lateral motion by being offset along the whole length, as reported in observations (Kuridze et al. 2015). An example of this are the features labeled F0 and F1.

Synthetic Hα line profiles associated with these features are either completely blue- or redshifted or they show an asymmetric, extended wing. These line profiles are caused by the vertical component of velocity with magnitudes larger than 30 − 40 km s⁻¹. In our model, these velocities appear mostly in the height range 2 − 4 Mm, at the line core formation height or slightly below it. By tracing magnetic field lines, we show that the vertical velocity that causes RBEs and RREs to appear in Hα line wings is always associated with the component of velocity perpendicular to the magnetic field line. Further analysis shows that features mainly outline the proxy location of Alfvén waves and, to a lesser degree, of fast magneto-acoustic waves. In some cases, proxies of all three wave modes can be found coinciding with the RBE and RRE formation. As most events were associated with Alfvén waves, the hypothesis that RBEs and RREs are signs of Alfvénic waves (Sekse et al. 2013a) is confirmed.

Tracing the magnetic field lines also demonstrates that strong flows at one of the footpoints perturbs field lines, as a result of which RBEs and RREs are formed. This explains why RBEs appear in groups (Yurchyshyn et al. 2013; Samanta et al. 2019). Strong flows would generate perturbations in all adjacent field lines, which would result in numerous shorter, thinner features, such as the ones modeled in this study. Our model, however, does not show many cases with strong jets. Likewise, the model generates more RREs than RBEs. This is inconsistent with observations. We find several possible reasons for this. Our detection method included many features that might not be counted as RREs if found in observations. Among these are those that resemble dynamic fibrils. These features, when observed, show 5-min oscillations in the region with a more inclined field (De Pontieu et al. 2007). Our synthetic Hα data set covers less than that, so there is a possibility that we caught most of them in their descending phase.

The resolution of our model can also be a reason why our generated RBE/RRE ratio is different than the observed one. Namely, the higher resolution would produce more turbulent flows, resulting in footpoint motions that could lead to more frequent magnetic reconnection. As the reconnection would happen near footpoints, the upflows would prevail in the height ranges where RBEs and RREs form and thus generate more RBEs than RREs. Also, flux recycling or small-scale emergence generated at higher resolution would result in fast lateral movement and would have the same effect. Although most of the flux emerging on granular scales gets pulled back (Danilovic et al. 2010), we do not exclude the scenario where these features are generated by the reconnection of an emerging field with a preexisting field, which our simulations do not model. Finally, the velocity magnitude increases with resolution, and hence more events will result in sufficient line shifts. The second possible reason for the mismatch of simulations and observations could be that we did not include the ion-neutral interaction effects, namely ambipolar diffusion (Martínez-Sykora et al. 2017). With ambipolar diffusion included, the plasma is not fully frozen in, so the jets may be more vertical and not follow the field lines. This would lead to a larger vertical velocity component and the identification of more features.

Apart from the unresolved cause of the mismatch between simulated and observed RBE/RRE ratios, a few more questions remain for future studies, including what determines the width of these features. In models such as the one from Srivastava et al. (2017) or in the cartoon by Sekse et al. (2013b), RBEs and RREs are generated at the edges of magnetic flux tubes. Based on this interpretation, observations (Srivastava et al. 2017; Shetye et al. 2021) would suggest that these flux tubes are resolved and sometimes as much as an arcsecond wide. Srivastava et al. (2021) further argue that if the waveguide is not spatially resolved, the opposing Doppler shifts will result in nonthermal broadening in optically thin lines while the chromospheric observables would sample only “the outermost shells of the structure”. We find no distinct structures as such in our model, although some synthetic RBE–RRE pairs can be traced to the same source. The model shows that while the Hα line core outlines density ridges (Leenaarts et al. 2012; Danilovic 2022), the formation of the line wings is more complex.

Finally, we did not discuss the photospheric source of these features in the model. Movies of traced field lines show a braiding of field lines close to a footpoint. This suggests the formation of a photospheric vortex flow. The vortex signatures are present in all areas of the simulation domain where the magnetic field is stronger. They are most prominent in the temperature maps of the upper photosphere, as shown by Moll et al. (2012). Vorticity caused by torsional motions in the photosphere can excite torsional Alfvén pulses that propagate along the magnetic field lines to the upper layers in the form of torsional Alfvén waves (Shelyag et al. 2013; Liu et al. 2019; Battaglia et al. 2021).

Movies

Movie 1 associated with Fig. 1 (fig1) Access here

Movie 2 associated with Fig. A.1 (F9) Access here

Movie 3 associated with Fig. A.1 (F01) Access here

Acknowledgments

This project has received funding from Swedish Research Council (2021-05613), Swedish National Space Agency (2021-00116) and the Knut and Alice Wallenberg Foundation. This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement No. 1852977. This research data leading to the results obtained has been supported by SOLARNET project that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no 824135. The calculations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC) at Linköping University and the PDC Centre for High Performance Computing (PDC-HPC) at the Royal Institute of Technology in Stockholm.

References

Appendix A: Supplementary m aterial

In addition the movie version of Fig. 1, we share two other movies, which show magnetic field lines for cases F9, F0, and F1. The field lines of F0 and F1 are plotted together, as shown in Fig. A.1, so that the initial overlapping of the magnetic field lines is visible.

Fig. A.1. Magnetic field lines traced for the RREs F0 (blue lines) and F1 (green lines). The panel on the left shows the top view with a vertical photospheric magnetic field in the background. Panels on the right show the projection of field lines in x (bottom) and y (top) planes with a density cut at the x,y coordinates of one of the initial seeds.

All Figures

	Fig. 1. Synthetic Hα images at different wavelength positions: the blue wing at Δv = −36 km s−1 (left), the nominal line center (middle), and the red wing at Δv = 36 km s−1 (right). The vertical blue line marks the position of the cut shown in Fig. 4. The movie is available online
In the text

	Fig. 2. Automatically identified synthetic RBEs and RREs. The panels show summed masks for RBEs and RREs, respectively. The black contours outline the unsigned photospheric vertical field of 1000 G.
In the text

	Fig. 3. Lifetime (left) and length (right) of tracked synthetic RBEs (black) and RREs (dashed red).
In the text

Fig. 4. Vertical cut along the blue vertical lines shown in Fig. 1. Vertical cuts in temperature, density, magnetic field strength, and vertical velocity are shown from top to bottom. The black line in all panels shows the τ = 1 at the nominal position of the Hα line center. Green and red contours mark τ = 1 levels at wavelength positions that correspond to Doppler shifts of Δv = −36 km s−1 and Δv = 36 km s−1, respectively.

In the text

Fig. 5. Hα four-panel diagram for the RBE at x = 10.7 Mm in Fig. 4. The plotted quantities, as functions of frequency and height, are indicated in the upper-left corner of each subplot and plotted in a grayscale. The vertical component of velocity as a function of height is shown in all panels (solid white line), with upward velocity being positive. The red line marks the τ = 1 position as a function of frequency and height. Planck and total source function are plotted in the upper-right panel with dotted and dashed lines, respectively. The line profile is shown with a solid white line in the lower-right panel.

In the text

	Fig. 6. Hα four-panel diagram for the RRE at x = 11.25 Mm in Fig. 4. The outline is the same as in Fig. 5.
In the text

	Fig. 7. Overview of the RBE–RRE sample chosen for magnetic field line tracing. For each of the 14 examples, a traced magnetic field line is plotted over the vertical component of the photospheric magnetic field. The diamonds mark the position of the starting seed used for line tracing: red marks RREs and blue RBEs. Yellow stars mark the position of the case F1.
In the text

	Fig. 8. Horizontal (y, solid line) and vertical (z, dashed line) displacement at constant x for four examples of RBEs and RREs. The vertical line marks the time when the seed is “planted”. The cases F0 and F1 are RREs and F6 and F9 RBEs. The labels are the same as in Fig. 7.
In the text

	Fig. 9. Evolution along a field line traced for four cases, labeled in the top-left corner. From left to right: total velocity, v, vertical velocity, vz, longitudinal velocity, vL, and perpendicular velocity, vP. Diamonds mark the location of starting seeds. The dotted lines follow a signal propagating with the Alfvén speed along the loop.
In the text

	Fig. 10. Proxies of different wave modes: Alfvén (left column), fast (middle column), and slow mode waves convolved with the contribution function at different wavelength positions: blue wing at Δv = −36 km s−1 (top row), red wing at Δv = 36 km s−1 (middle row), and nominal line center (bottom row). The blue vertical line marks the position of the cut shown in Fig. 4.
In the text

	Fig. A.1. Magnetic field lines traced for the RREs F0 (blue lines) and F1 (green lines). The panel on the left shows the top view with a vertical photospheric magnetic field in the background. Panels on the right show the projection of field lines in x (bottom) and y (top) planes with a density cut at the x,y coordinates of one of the initial seeds.
In the text